How to Factorize 2√2x^3 - 343 Using Advanced Algebraic Techniques

The polynomial 2√2x^3 - 343 can be factorized using algebraic techniques, including understanding complex roots of unity and standard factorization methods. Let's go through the process step-by-step to grasp how we can tackle this problem.

Step 1: Initial Simplification

Begin with the given polynomial:

2√2x^3 - 343 0

This can be simplified as:

2√2[x^3 - 343/2√2] 0

Let's denote this as px 2√2x^3 - 343. We can rewrite it using the difference of cubes formula, which states that a^3 - b^3 (a - b)(a^2 ab b^2).

Step 2: Applying Difference of Cubes Formula

Define:

a 7/√2 μ 2√2

The polynomial can be rewritten as:

μ[x^3 - (7/√2)^3]

Using the difference of cubes formula, we get:

μ[x - a][x^2 ax a^2]

Substituting back the values of μ and a, we obtain:

(2√2)[x - 7/√2][x^2 (7/√2)x (7/√2)^2]

Further Breakdown

This can be further simplified by recognizing that the terms inside the second set of brackets can be simplified. Let's rewrite the expression in terms of complex cube roots of unity:

2√2[x - 7/√2][x - 7/√2 w][x - 7/√2 w^2]

Where:

w 1/2 {-1, i√3} are the complex cube roots of unity.

Key Concepts and Terminologies

Factorization: The process of breaking down a polynomial into its constituent factors. In this case, we used the difference of cubes formula and complex roots to achieve the factorization.

Cubic Polynomial: A polynomial of degree three, such as 2√2x^3 - 343. Understanding how to factorize such polynomials is essential in algebra.

Algebraic Techniques: Various methods used to solve algebraic equations and simplify expressions. In this problem, we used the difference of cubes and the properties of complex numbers.

Finding the Factors

By following the steps above, we determined that the factors of the polynomial 2√2x^3 - 343 are:

(2√2)[x - 7/√2][x - 7/√2 w][x - 7/√2 w^2]

To verify, you can expand this expression back and confirm that it matches the original polynomial.

Conclusion

Factorizing polynomials, especially cubic ones, can be complex but follows a structured approach. Understanding the difference of cubes formula and the properties of complex numbers is key. By applying these techniques, you can solve more complex algebraic problems and factorize polynomials step-by-step.

Frequently Asked Questions (FAQ)

Q: What is a cubic polynomial?

A cubic polynomial is a polynomial of degree three, typically written as ax^3 bx^2 cx d. In this problem, we dealt with the specific form of a cubic polynomial: 2√2x^3 - 343.

Q: How do complex roots of unity help in factorization?

Complex roots of unity, especially in the context of polynomials, provide a way to factorize polynomials that have roots that are not real numbers. In the case of 2√2x^3 - 343, the roots involve complex numbers, making use of the properties of w (a complex cube root of unity) essential.